set point tracking of ball and beam system using genetic

NUST Publishing, © (2018), ISSN: 2070-9900

Set Point Tracking of Ball and Beam System using Genetic Algorithm based PI-PD Controller

Tayyab Ali*, Suheel Abdullah Malik, Muhammad Adeel, Muhammad Amir, Department of Electrical Engineering

International Islamic University

Islamabad, Pakistan *Corresponding author: [email protected]

Abstract The ball and beam system is one of the commonly used benchmark control apparatus for evaluating numerous different real

systems and control strategies. It is inherently nonlinear and open loop unstable system. In this paper, we have suggested

Evolutionary Algorithm (EA) based Proportional Integral-Proportional Derivative (PI-PD) controller for the set point

tracking of this well-known ball and beam system. A linearized model of the ball and beam system is deduced and PI-PID

control methodology is employed. The popular EA technique such as Genetic algorithm (GA) is used for tuning of the

controller. The optimized values of the controller parameters are achieved by solving a fitness function using GA. The

transient performance of the proposed GA based PI-PD controller (GA-PI-PD) is evaluated by carrying set point tracking

analysis of the ball and beam system through MATLAB/Simulink simulations. Furthermore, the performance of GA-PI-PD

controller is investigated using four different performance indices such as Integral of squared value of error (ISE), Integral of

time multiplied by squared value of error (ITSE), Integral of absolute value of error (IAE) and Integral of time multiplied by

absolute value of error (ITAE). The comparison of transient performance including rise time, settling time and % overshoot

is made with SIMC-PID and H-infinity controllers. The comparison reveals that GA-PI-PD controller yielded transient

response with small % overshoot and settling time. The superior performance of the GA-PI-PD controller has witnessed that

it is highly effective for maintain good stability and the set point tracking of ball and beam system with fast settling time and

less overshoot than SIMC-PID and H-infinity controllers.

Keywords: Stability control, Ball and Beam system, Set point tracking, Evolutionary Algorithms, Genetic Algorithm

(GA), Proportional Integral-Proportional Derivative (PI-PD) controller, SIMC based PID controller, H-infinity controller,

Open loop unstable systems.

Introduction The ball & beam system has been used to study and

analyze the stability control of many control

engineering problems [1]. Ball & beam system

comprises of ball, beam, gear, motor and position

sensors. The ball and beam system is an open-loop

unstable system [2]. In ball & beam system, main aim is

to change the ball position as desired by varying the

beam angle [3-4]. Since open loop response of this

system is unstable, so a controller is always required to

make this system stable. For this purpose, many

controllers have been used for the stabilization of ball

and beam system such as Fuzzy controller, SIMC-PID,

H-infinity and LQR and [2-11]. Many traditional tuning

techniques have been used for the tuning of these

controllers. Evolutionary computational techniques

such as Differential Evolution (DE) and Particle Swarm

Optimization (PSO) have been explored successfully

for the tuning of controllers. It has been observed that

these evolutionary computational techniques have given

satisfactory transient response than classical tuning

techniques. In [8], Differential Evolution based PID

(DE-PID) controller has been implemented for the

control ball & beam system. In [9], I-PD controller

tuned by Particle Swarm Optimization (PSO) has been

designed to control the desired system. In this work, PI-

PD controller has been used for the set point tracking of

ball & beam system. Genetic Algorithm (GA) is an

evolutionary computational technique, which has been

utilized in this research for the tuning of proposed

controller. The comparison of the GA-PI-PD controller

with SIMC based PID (SIMC-PID) and H-infinity

controller is presented to evaluate the working of the

proposed controller.

Mathematical Modeling and Controller Design

Transfer function of ball and beam system between gear

angle (𝛉(s)) and ball position (P(s)) has been derived in

this section. The schematic diagram of the system is

provided in Figure 1. Parameters of the ball & beam

system and their values are specified in Table 1.

Fig 1: Schematic of Ball and Beam System

The linear acceleration of the ball along the mounted

beam is written by using Lagrangian equation of

motion[2,9].

(J

R2 + m ) �̈� + mgsinα − m𝑝�̇�2 (1)

where

R is representing the ball’s radius

m is representing the mass of the ball

g is representing the gravitational acceleration

J is representing the moment of inertia of the ball

p is representing the ball’s position http://dx.doi.org/10.24949njes.v11i1.287

mailto:[email protected]


α is defined as the beam angle

Linearization of equation (1) about 𝛼 = 0 can be

written as:

(J

R2 + m ) �̈� = − mgα (2)

also

α = (𝑑

L) θ (3)

where,

L is representing the beam’s length

d is the distance between joint of the lever arm and the

center of the gear

Substituting equation (3) into equation (2),

(J

R2 + m ) �̈� = − mg (𝑑

L) θ (4)

Taking Laplace transform of equation (4) yields as,

GBB(s) =P(s)

θ(s)= −

mg𝑑

L(J

R2+m)

1

𝑠2 (5)

GBB(s) represents the transfer function of the ball and

beam system.

Table 1: Ball and Beam system’s parameters

After substitution of parameters’ values in equation 5

from Table 1, GBB(s) can be written as

GBB(s) = P(s)

θ(s)=

0.7

s2 (6)

Controllers for Ball and Beam System

In this research, Genetic Algorithm (GA) based

Proportional Integral-Proportional Derivative (PI-PD)

controller has been used for set point tracking of the

ball & beam system. PI-PD controller has two degrees

of freedom. It has both close loop as well as feedback

characteristics, which lead to an efficient set point

tracking of the desired system [12-13].

PI-PD Controller

In PI-PD controller, Proportional Derivative (PD)

controller is associated with feedback path whereas

Proportional Integral (PI) controller is associated with

feed-forward path. Figure 2 shows the block diagram

representation of PI-PD controller. Y(s) represents the

output of the system whereas R(s) denotes the input of

the system.

Fig 2: Block diagram of PI-PD controller along with ball

and beam system

Close-loop transfer function of the proposed PI-PD

controller can be obtained as (GC(s)),

GC(s) =Y(s)

R(s)=

0.7KPs+0.7Ki

s3+0.7s2+1.4Kps+0.7Ki (7)

Controller’s Tuning

To determine optimum value of the controller’s

constants (KP, Ki, KD), one has to utilize different

tuning techniques. Many conventional tuning

techniques have been used to do this but in recent past

evolutionary computational techniques have been

widely used to tune the controllers. These techniques

include Particle Swam Optimization (PSO) and

Differential Evolution (DE) etc. Genetic Algorithm

(GA) has been explored in this research work for tuning

purpose. The block diagram representation for the set

point tracking of ball & beam system with evolutionary

computation based controller is demonstrated in Figure

3.

Fig 3: Block Diagram of the set point tacking of ball

and beam system

Genetic Algorithm (GA) Genetic Algorithm (GA) is as evolutionary algorithm

based upon Darwin’s theory. GA was invented in the

early 1970’s by John Holland. It is used for the solution

of different optimization problems [14]. GA is a

stochastic algorithm, which has been provoked by

evolutionary genetics [15]. The basic idea is to create

new random generations till you will find out best

solution. After getting optimum solution of a problem,

algorithm has been stopped. The flow chart diagram of

the GA is given in Figure 4.

Parameter’s

symbol

Description of

Parameters

Value

L Beam’s length 0.4 m

R Radius of the Ball 0.015 m

d Offset of the lever

arm

0.04 m

J Ball’s moment Inertia

of the

2mR2/5

kgm^2

g Gravitational

acceleration

-9.8m/s2

m Mass of the ball 0.11 kg

13 13


Fig 4: Flow Chart of Genetic Algorithm (GA)

In order to find out optimum value of unknown

parameters, an objective function is always required to

be minimized. For this purpose, error signal e(t) is

taken as a objective function (fitness function).

System’s performance can be calculated by using

different performance indices, which are associated

with different parameters like overshoot (os), settling

time (ts), steady state error (ess.) and rise time (tr) etc.

Four different types of errors including Integral of

squared value of error (ISE), Integral of time multiplied

by squared value of error (ITSE), Integral of absolute

value of error (IAE) and Integral of time multiplied by

absolute value of error (ITAE) have been considered as

performance indices in this research work. These

indices have been minimized using Genetic Algorithm

(GA), which is implemented through MATLAB

Optimization Toolbox. These performance indices can

be obtained as,

ISE = ∫ e2(t)dtT

0 (8)

IAE = ∫ |e(t)|dtT

0 (9)

ITAE = ∫ t|e(t)|dtT

0 (10)

ITSE = ∫ te2(t)dtT

0 (11)

Simulation Results and Discussion

In this section, GA-PI-PD controller is implemented for

the set point tracking of ball and beam system given by

equation 6.

Table 2: PI-PD controller’s tuning using Genetic

Algorithm (GA)

MATLAB/Simulink has been utilized for simulation

purpose. In all simulations, different set points (10cm,

20cm and 30cm) are taken as reference positions.

Figure 5 shows the open loop set point tracking of the

system. It can be observed that the response is growing

with time i.e. an unstable response.

Fig 5: Open-loop set point tracking of ball and beam

system

Implementation of GA-PI-PD controller for the set point

tracking is provided in this section. In Table 2, optimum

tuning parameters of PI-PD controller acquired by GA have

been provided. Figure 6 and 7 shows the output responses

of GA-PI-PD controller for IAE and ISE respectively.

Similarly Figure 8 and 9 shows the output responses of GA-

PI-PD controller for ITSE and ITAE respectively. Table 3

represents the transient response performance of GA-PI-PD

controller with each performance index. It can be observed

from the results of Table 3 that GA-PI-PD controller with

ISE exhibits relatively less rising time with zero %

overshoot. Moreover, it can be observed that GA-PI-PD

controller with ITAE gives lower value of settling time.

Finally, it can be concluded that GA-PI-PD controller yields

much better transient response in terms of steady state error

(ess.), settling time (ts), overshoot (OS) and rise time (tr).

Gain

Parameters

Performance Index

KP Ki KD

ISE 17.261 29.99 4.96

ITSE 14.19 29.99 6.68

IAE 13.99 29.99 7.55

ITAE 14.46 29.99 9.21

14


Fig 6: Set point tracking of GA-PI-PD

controller with IAE


controller with ISE


controller with ITSE


controller with ITAE

Table 3: Comparison of GA-PI-PD controller with

different performance indices

Figure 10 shows the comparison of GA-PI-PD controller with

SIMC-PID and H-infinity controller as used in [2]. The

performance comparison is also provided in Table 4. Figure

10 and Table 4 reveal that GA-PI-PD controller yields very

negligible % overshoot with rise time (tr) and settling time (tS)

relatively less than SIMC-PID and H-infinity controller.

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

Time (sec)

Ba

ll p

os

itio

n (

cm

)

Desired Position

IAE (GA based PI-PD controller)

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

Time (sec)

Ball p

osit

ion

(cm

)

Desired Position

ISE(GA based PI-PD controller)

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

30

35

Time (sec)

Ba

ll p

os

itio

n (

cm

)

Desired Position

ITSE (GA based PI-PD controller)

Performance

Parameter

Performance

Index

Settling

Time

(sec)

s-s

Erro

r

Rise

Time

(sec)

%

Overshoo

t

IAE 2.30 0 0.60 1.73

ITAE 1.08 0 0.70 1

ISE 3.24 0 0.46 0

ITSE 2.33 0 0.55 1.90

15


Fig 10: Set point tracking of PI-PD controller

vs. SIMC based PID and H-infinity controller

Table 4: Comparison of GA-PI-PD with SIMC-PID

and H-infinity controller [2]

Performance

Parameter

Controller

Settling

Time

(sec)

s-s

Erro

r

%

Oversho

ot

Rise

Time

(sec)

GA-PI-PD

ISE 3.24 0 0 0.46

ITAE 1.08 0 1 0.70

SIMC-PID controller 10.9 0 40 1

H-infinity controller 3.7 0 6.7 1.1

Conclusions

The set point trancking response of ball and beam

system has been investigated using PI-PD controller.

An evolutioary computational technique Genetic

Algorithm (GA) has been utilized to find out optimum

parameters of the proposed controller. ITSE, ISE, ITAE

and IAE have been utilized for the assessment of PI-PD

controller. Simulation results reveal that GA-PI-PD

controller with each performance index is very effective

as comapred to SIMC-PID and H-infinty controller [2].

GA-PI-PD controller has very little overshoot (os),

settling time(tS) and rise time (tr). Since GA-PI-PD

controller has very small % overshoot, it will be very

reliable for the plant dynamics. Similarly rise time and

settling time have been reduced so that sytem will

achieve its desired position within very small time

intrerval. Finally it can be concluded that GA-PI-PD

controller is much efficient and valuable for the set

point tracking of ball & beam systems which will be

very supportive for the control of many engineering

problems in the future.

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set point tracking of ball and beam system using genetic

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